Theorem f1cnv 3672

Description: Two ways to express that a set A (not necessarily a function) is one-to-one. Each side is equivalent to Definition 6.4(3) of [TakeutiZaring] p. 24, who use the notation "Un₂ (A)" for one-to-one. We do not introduce a separate notation since we rarely use it.

Assertion

Ref	Expression
f1cnv	|- (`'`'A:dom A-1-1->V <-> (Fun `'A /\ Fun `'`'A))

Proof of Theorem f1cnv

Step	Hyp	Ref	Expression
1		df-f1 3201	. 2 |- (`'`'A:dom A-1-1->V <-> (`'`'A:dom A-->V /\ Fun `'`'`'A))
2		fnf 3634	. . . 4 |- (`'`'A Fn dom A <-> `'`'A:dom A-->V)
3		df-fn 3199	. . . . 5 |- (`'`'A Fn dom A <-> (Fun `'`'A /\ dom `'`'A = dom A))
4		dmcnvcnv 3342	. . . . 5 |- dom `'`'A = dom A
5	3, 4	mpbiran2 731	. . . 4 |- (`'`'A Fn dom A <-> Fun `'`'A)
6	2, 5	bitr3 175	. . 3 |- (`'`'A:dom A-->V <-> Fun `'`'A)
7		relcnv 3441	. . . . 5 |- Rel `'A
8		dfrel2 3491	. . . . 5 |- (Rel `'A <-> `'`'`'A = `'A)
9	7, 8	mpbi 189	. . . 4 |- `'`'`'A = `'A
10		funeq 3541	. . . 4 |- (`'`'`'A = `'A -> (Fun `'`'`'A <-> Fun `'A))
11	9, 10	ax-mp 7	. . 3 |- (Fun `'`'`'A <-> Fun `'A)
12	6, 11	anbi12i 484	. 2 |- ((`'`'A:dom A-->V /\ Fun `'`'`'A) <-> (Fun `'`'A /\ Fun `'A))
13		ancom 437	. 2 |- ((Fun `'`'A /\ Fun `'A) <-> (Fun `'A /\ Fun `'`'A))
14	1, 12, 13	3bitr 177	1 |- (`'`'A:dom A-1-1->V <-> (Fun `'A /\ Fun `'`'A))

Syntax hints:   <-> wb 146   /\ wa 223   = wceq 958  Vcvv 1814  `'ccnv 3175  dom cdm 3176  Rel wrel 3181  Fun wfun 3182   Fn wfn 3183  -->wf 3184  -1-1->wf1 3185

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-fun 3198  df-fn 3199  df-f 3200  df-f1 3201

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